Tangents are drawn to a unit circle with centre at the origin from each point on the line x + y = 2 . Then, the area (in sq. units) of the locus of the mid-point of the chord of contact is

Step-by-step solution

x 1 , y 1 lies on x + y = 2 ⇒ x 1 + y 1 = 2 … i Chord of contact with respect to x 1 , y 1 x x 1 + y y 1 = 1 Also equation of chord whose mid point is h , k h 2 + k 2 = h x + k y ∴ x 1 h = y 1 k = 1 h 2 + k 2 ⇒ x 1 = h h 2 + k 2 , y 1 = k h 2 + k 2 Substituting in i , we get, h h 2 + k 2 + k h 2 + k 2 = 2 Hence, the required locus is x 2 + y 2 = x 2 + y 2 The area of the circle is π 8 sq. units